ARTICLE IN PRESS
Journal of Biomechanics 37 (2004) 827–835
Young Scientist Pre-Doctoral Award
Mechanical and metabolic requirements for active lateral stabilization
in human walking
J. Maxwell Donelana,*, David W. Shipmanb, Rodger Kramc, Arthur D. Kuod
a
Department of Physiology, University of Alberta, Edmonton, AB T6G 2H7, Canada
Department of Integrative Biology, University of California, Berkeley, CA 94720-3140, USA
c
Department of Integrative Physiology, University of Colorado, Boulder, CO 80309-0354, USA
d
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA
b
Received 16 June 2003; accepted 23 June 2003
Abstract
Walking appears to be passively unstable in the lateral direction, requiring active feedback control for stability. The central
nervous system may control stability by adjusting medio-lateral foot placement, but potentially with a metabolic cost. This cost
increases with narrow steps and may affect the preferred step width. We hypothesized that external stabilization of the body would
reduce the active control needed, thereby decreasing metabolic cost and preferred step width. To test these hypotheses, we provided
external lateral stabilization, using springs pulling bilaterally from the waist, to human subjects walking on a force treadmill at
1.25 m/s. Ten subjects walked, with and without stabilization, at a prescribed step width of zero and also at their preferred step
width. We measured metabolic cost using indirect calorimetry, and step width from force treadmill data. We found that at the
prescribed zero step width, external stabilization resulted in a 33% decrease in step width variability (root-mean-square) and a 9.2%
decrease in metabolic cost. In the preferred step width conditions, external stabilization caused subjects to prefer a 47% narrower
step width, with a 32% decrease in step width variability and a 5.7% decrease in metabolic cost. These results suggest that (a) human
walking requires active lateral stabilization, (b) body lateral motion is partially stabilized via medio-lateral foot placement, (c) active
stabilization exacts a modest metabolic cost, and (d) humans avoid narrow step widths because they are less stable.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Biomechanics; Biped; Energetics; Locomotion; Gait
1. Introduction
The dynamic stability of walking depends on the
interaction between the passive dynamics of the
musculoskeletal system and active control produced by
the central nervous system (CNS). A passively stable
musculoskeletal system is able to reject small perturbations without direct intervention by the CNS. But
instability in the passive dynamics must be countered
by active control, which requires coordinated sensing of
perturbations, generation of appropriate response motor programs, and production of compensatory body
motions. This places a computational demand on the
CNS and a mechanical and metabolic demand on
muscles. Active control is also subject to limitations in
the precision of sensory receptors and muscles. There
*Corresponding author.
E-mail address: mdonelan@ualberta.ca (J.M. Donelan).
0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jbiomech.2003.06.002
may therefore be significant advantages to harnessing
any passive stability afforded by walking in order to
minimize active control and its associated costs.
Mathematical modeling suggests that the dynamics of
bipedal walking is passively stable, with the exception of
lateral motion. McGeer (1990) demonstrated that with
little actuation and no active control, stable walking
gaits can arise from the dynamic interactions between
the legs and the ground. These gaits, when restricted to
the sagittal plane, are passively stable. But passive
dynamic walking models that move side-to-side and
have human-like mass distribution are laterally unstable, even though they retain stability within the
sagittal plane (Kuo, 1999). Applying these principles to
human walking suggests a need for dedicated active
feedback control of lateral motion (Fig. 1a). Kuo (1999)
showed that medio-lateral foot placement is a particularly effective method for stabilizing lateral balance
in passive walking models. Subsequent empirical
ARTICLE IN PRESS
J.M. Donelan et al. / Journal of Biomechanics 37 (2004) 827–835
828
Without External Lateral Stabilization
gait dynamics
CNS
Control
Lateral
Stance/Swing
lateral
motion
sagittal
plane
motion
passive stability
active feedback
(a)
With External Lateral Stabilization
external stabilization
CNS
Control
Lateral
Stance/Swing
lateral
motion
sagittal
plane
motion
passive stability
active feedback
(b)
Fig. 1. The dynamic stability of walking depends on the interaction
between the passive dynamics of the musculoskeletal system and active
control produced by the CNS. (a) In normal walking, ‘‘passive’’
feedback is available to stabilize motion in the sagittal plane, but stable
lateral motion requires ‘‘active’’ feedback and active control by the
CNS. (The solid lines denote pathways that are present or necessary.
The dashed lines represent pathways that are not present or not
necessary.) (b) External lateral stabilization provides passive feedback
control to stabilize lateral motion making active feedback control no
longer necessary.
measurements demonstrated that humans have increased medio-lateral foot placement variability when
walking with reduced visual information (Bauby and
Kuo, 2000). This suggests that humans actively stabilize
lateral motion using medio-lateral foot placement and
that this active control mechanism is less precise when
there is less sensory information. If subjects were
passively stable, they would be expected to have no
sensitivity to sensory information.
Active stabilization of lateral balance in human
walking may exact a significant metabolic cost for two
reasons. First, active movement of the legs to adjust
medio-lateral foot placement exacts a metabolic cost
(Donelan et al., 2001; Shipman et al., 2002). Second,
mechanical work is performed to redirect the center of
mass (COM) during the transition between steps
(referred to as step-to-step transition costs, Donelan
et al., 2002). This work exacts a proportional metabolic
cost and increases with the square of step width
(Donelan et al., 2001). The consequence of this nonlinear dependence upon step width is that, for the same
average step width, variability in medio-lateral foot
placement will increase the average step-to-step transition cost.
Lateral instability may play a role in determining the
step width preferred by humans. The mechanical work
of step-to-step transitions is minimized when humans
walk with zero step width (Donelan et al., 2001). But the
step width that minimizes metabolic cost—and is in fact
preferred by humans—is about 0.13L (where L is leg
length). Other factors must therefore increase metabolic
cost for narrow step widths. One possibility is that
walking with narrow steps is more unstable, requiring
greater active control. This intuitive prediction is
supported by mathematical modeling (Kuo, 1999). The
trade-off between minimizing mechanical work to
redirect the COM and the cost of active stabilization
of lateral balance may determine the step width
preferred by humans.
We studied the mechanical and metabolic requirements of active lateral stabilization in human walking. A
mathematical model, based on passive dynamics, predicts that walking could be completely stabilized using
only external lateral forces (Fig. 2), thus removing the
need for active control. Inspired by this model, we built
a device that applied lateral stabilizing forces to human
subjects as they walked on a force treadmill (Fig. 3). We
hypothesized that external lateral stabilization would
reduce the amount of active stabilization required to
walk (Fig. 1b), resulting in less variability of mediolateral foot placement and a reduced metabolic cost. We
also hypothesized that when subjects were allowed to
choose their step width, they would prefer a narrower
step width when externally stabilized, because the
advantages of wider steps for active stabilization are
eliminated if walking is made passively stable.
2. Methods
2.1. Model predictions
A previously developed mathematical model, based
on passive dynamics, predicts that walking is passively
unstable in the lateral direction (Donelan et al., 2001;
Kuo, 1999). Passive dynamic walking refers to a class of
walking models that walk with little or no actuation and
no active control. During single support phases, the legs
act as freely swinging, coupled pendula. Double support
phases function as transitions between single support
phases and are modeled as instantaneous, inelastic
collisions between foot and ground. The energy lost
during each transition can be replaced by a toe-off
impulse applied along the trailing leg, or by walking
down a gentle slope (Kuo, 1999; McGeer, 1990). We
previously studied stability in a simple three-dimensional model (Fig. 2a), with leg inertia condensed into
point masses at the pelvis and feet, and the mass of the
feet much smaller than that of the pelvis (Donelan et al.,
2001; Garcia et al., 1998). The model predicted that
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J.M. Donelan et al. / Journal of Biomechanics 37 (2004) 827–835
3-D Walking Model
K
Stability analysis
40
φ
ψ
2β
swing swing
leg
leg
m
Front
stance
leg
Eigenvalue magnitude
M
θ
stance
leg
20
stable zone
m
0
0.0
Front view
829
Side view
0.5
1.0
1.5
2.0
Stabilizer Stiffness (K)
(b)
(a)
Fig. 2. (a) Simple three-dimensional walking model with external lateral stabilization. Model has three degrees of freedom: stance (y) and swing (f
leg angles in the sagittal plane, and roll angle (c)). Mass is concentrated in points located at the pelvis (M) and feet (m). Step width is adjusted by
changing the splay angle, b: External lateral stabilization is provided by a spring with stiffness, K: (b) Stability analysis (see Appendix A)
demonstrates that increasing stabilizer stiffness, K; decreases instability. Without external stabilizing forces, the walking model is very unstable (yintercept is much greater than 1). For normalized stiffness K > 1; the walking model is passively stable. Stability is described by the eigenvalue
corresponding to the unstable lateral mode in the step-to-step dynamics, with a magnitude less than one forming the boundary of the stable zone. For
all values of K; the least stable mode was for lateral motion.
Force
transducer
Spring
Force
transducer
Spring
8.5m
Hip belt
8.5m
Treadmill
Force platform
Fig. 3. External lateral stabilizer. Two lightweight elastic cords
attached to a padded waist belt stabilized lateral motion by pulling
laterally on walking subjects.
walking is passively stable in the plane of progression
but has one highly unstable mode (see current Appendix
for details), confined mainly to the lateral motion.
Active feedback control can be used to stabilize this
mode, for example, through medio-lateral adjustments
to foot placement (Bauby and Kuo, 2000).
In the present study, we found that the external
application of spring-like lateral forces can passively
stabilize our walking model. We modified our previously
developed model to include a purely lateral elastic
restoring force, F ; applied to the pelvis
F ¼ KL sin c;
ð1:1Þ
where K is the spring stiffness, L the leg length, and c
the roll angle (Fig. 1a). The effect of the stabilizing
spring is demonstrated most clearly when the model
walks at zero step width, and F merely acts as
proportional control about a pelvis trajectory that has
no lateral motion. In the absence of disturbances, the
spring exerts no force and has no effect on the walking
model; it only counters disturbances out of the plane of
progression. We studied the stability of this model by
examining its response to small perturbations: a matrix
describing the step-to-step dynamics will have eigenvalues with magnitude less than 1 for a stable system (see
Appendix A). We found that increasing K acts to
stabilize the model (Fig. 2b), reducing the need for active
feedback control. For all values of K; the least stable
mode was almost entirely associated with lateral motion;
the lateral components accounted for more than 99.9%
of the length of the eigenvector of the least stable mode.
Similar predictions apply for more complex models
and situations. The passive lateral instability holds for a
wide range of step widths, although narrow steps are
slightly more unstable (Kuo, 1999; McGeer, 1990). We
found that the stabilizing effect of the external force
from Eq. (1.1) also applies to non-zero step widths
where the pelvis moves side to side. We also investigated
the effect of a more realistic stabilizing spring by adding
a damping term to Eq. (1.1). A small amount of
damping results in stability at a smaller K; but at the
cost of increased energy dissipation when the pelvis
moves laterally. Finally, we found that these stability
properties are retained in a more complex model with
realistic anthropomorphic parameters (Kuo, 1999).
2.2. Experimental procedures
We compared the mechanics and metabolic cost in 10
human adult subjects, walking on a treadmill with and
ARTICLE IN PRESS
J.M. Donelan et al. / Journal of Biomechanics 37 (2004) 827–835
830
without externally applied stabilization. All volunteer
subjects (six male, four female, body mass 73.3714.2 kg;
leg length 0.9570.05 m; mean7s.d.) were healthy and
exhibited no clinical gait abnormalities. Before the
experiments began, subjects gave their informed consent
to participate in accordance with University of California policy. The equipment consisted of an external
lateral stabilizer, a force treadmill and an indirect
calorimetry system. The experimental design required
the subjects to walk at a prescribed zero step width and
then at their preferred step width, both with and without
external lateral stabilization—a total of four walking
trials for each subject.
The external lateral stabilizer consisted of two lightweight elastic cords that attached to the subject and
pulled in both lateral directions (Fig. 3). Each adjustable
cord was made of 8.0 m of nylon cord in series with
0.5 m of rubber tubing. One end of each cord was
attached to a force transducer, used to monitor
stabilizing forces, and the force transducer was anchored
to a wall. The other end was attached to each subject via
a padded belt (taken from a hip-supported backpack)
worn securely about the waist. The two cords had an
effective spring constant of approximately 1700 N/m
and negligible damping (14 N-s/m), identified by oscillating a known mass between the cords and estimating
the parameters from a second-order damped oscillator
model. We used long length cords to ensure that any
non-lateral forces exerted by the apparatus were
negligible. For example, peak vertical displacements of
the COM are approximately 0.04 m resulting in vertical
forces of less than 0.1% body weight. We also estimated
unwanted fore-aft forces to be less than 0.1% body
weight.
We used the center-of-pressure of ground reaction
forces (see Fig. 4) to measure average step width and
step length, as well as variability in step width and step
length. We calculated instantaneous center of pressure
using ground reaction forces and moments measured
with the force treadmill (Kram et al., 1998). The ground
reaction forces and moments were sampled at 100 Hz
and low-pass filtered with a 25 Hz cut-off frequency
(fourth-order, zero-phase-shift Butterworth digital filter). Step width was defined as the lateral distance
between consecutive centers of pressure for the time
sample immediately before heel-strike. We choose to use
this time sample because the ground reaction forces and
moments are large during this period of single support,
thereby maximizing the signal-to-noise ratio. Step length
was defined as the fore-aft distance between consecutive
centers of pressure summed with the distance traveled
during the step (the latter term is necessary to correct for
treadmill belt speed). The distance traveled during each
step was calculated from the product of treadmill
velocity and the time between consecutive heel-strikes.
We identified heel-strikes from the characteristic forward translation of the center of pressure that occurs at
the beginning of double support. We calculated each
subject’s instantaneous center of pressure, and the mean
and root-mean-square (rms) of step width and step
length, for 200 steps after the subject had been walking
for 150 s. To account for differences in body size, we
normalized all length measurements by leg length. The
precision of our estimates of step width and step length
was limited by the precision by which centers of pressure
and timing of gait events could be detected. Compared
to more direct kinematic methods of estimating these
same measures (Bauby and Kuo, 2000), the force plate
method is slightly less precise, but nonetheless adequate
for detecting changes in step variability.
We determined the COM position (see Fig. 4) using
the vector sum of the external forces acting on the
subject. In our experiment, the only substantial external
forces acting on the subject are from gravity, the ground
and the external lateral stabilizer. These were measured
using the mass of the subject, the force treadmill,
and force transducers in series with the stabilizing
cords, respectively. We calculated COM acceleration by
Condition: Prescribed Step Width
COM  without external stabilization
COP 
COM  with external stabilization
COP 
(a)
Condition: Preferred Step Width
step
width
0.2m
0.2m
step length
(b)
Fig. 4. Center of pressure (COP) and COM trajectories for walking (a) at the prescribed zero step width, and (b) at the preferred step width. Data
shown are averages of 200 steps from each subject walking with and without external lateral stabilization. Our definitions of step width and length are
illustrated for the condition of walking at preferred step width, without external lateral stabilization. Foot placement varied from step to step, relative
to the average step width and length. The scale is the same for both (a) and (b).
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J.M. Donelan et al. / Journal of Biomechanics 37 (2004) 827–835
dividing the vector sum of the external forces by body
mass. COM position is the second time-integral of the
acceleration subject to appropriate boundary conditions
(Cavagna, 1975; Donelan et al., 2002). Applying a high
pass filter, with a cut-off frequency of one-tenth of each
subject’s step frequency, after each integration removed
the effect of any low frequency bias (fourth-order, zerophase-shift Butterworth digital filter). We performed
each integration over the same 200 steps used for center
of pressure calculations. Using a large number of steps
increases the accuracy of our boundary condition
assumptions.
We used indirect calorimetry to measure metabolic
cost (McLean and Tobin, 1987). We measured the rates
of oxygen consumption ðV’ O2 Þ and carbon dioxide
production ðV’ CO2 Þ using an open circuit respirometry
system (Physio–Dyne Instrument Co., Quogue, NY).
After a resting trial in which subjects stood quietly, we
conducted the remaining walking trials in random order.
Each trial lasted seven minutes, including three minutes
for the subjects to reach steady state (Poole and
Richardson, 1997) and then three minutes for collection
of average V’ O2 (ml O2 s1) and V’ CO2 (ml CO2 s1) data.
We discarded the seventh minute of data due to a delay
between breath expiration and breath analysis. We
calculated metabolic power (W) for each trial using the
following standard equation (Brockway, 1987):
s ’
Pmet; gross ¼ 16:58mlWOs 2 V’ O2 þ 4:51mlWCO
VCO2 :
2
ð1:2Þ
We subtracted the metabolic power for standing from
all walking values and divided by body mass to derive
net metabolic power (W/kg). We also calculated
dimensionless net metabolic cost of transport (COT)
by dividing net metabolic power by the gravitational
constant (9.81 m/s2) and by walking velocity (m/s). The
metabolic measurements were collected during the same
trial as the mechanics measurements.
We measured the effects of external lateral stabilization as subjects walked at both a prescribed zero step
width and at their preferred step width. In all conditions, subjects walked at a fixed treadmill speed of
1.25 m/s, with average step frequency and step length
controlled by instructing subjects to match a metronome
set to their previously determined preferred step
frequency. The arms were crossed in all conditions,
both to control against changes in arm motion, and
because the external stabilizer interferes with normal
arm swinging. The prescribed step width conditions
were designed to detect changes in the metabolic cost of
active control induced by external stabilization, independent of other potential metabolic costs that might
change with step width. We chose a target of zero step
width because our device stabilizes the body about a
trajectory with zero lateral movement. Zero average step
width was controlled by instructing subjects to walk on
a single line marked on the treadmill belt. This condition
831
was repeated with and without external stabilization.
We then used a separate set of conditions, in which
subjects walked at their preferred (i.e. freely selected)
step width, to determine whether external stabilization
affects preferred step width. Subjects practiced all
conditions prior to data collection.
We performed our statistical comparisons using
paired t-tests with a level of significance of po0.05.
Reported p values less than 0.05 are from one-tailed
tests and p values greater than 0.05 are from two-tailed
t-tests.
3. Results
For both the prescribed step width and preferred step
width control trials (i.e. without external stabilization),
subjects walked with a preferred step frequency of
1.7670.10 Hz (mean7s.d.), and a preferred step length
of 0.75670.052L (where L is leg length). In the
prescribed step width control condition, subjects walked
with a step width of 0.01170.009L (Fig. 5), close to the
target width. Foot placement variabilities, described by
rms step width and step length, were 0.01670.004L and
0.029 0.007L, respectively. Peak-to-peak lateral displacement of the COM was 0.00570.003L. The net
metabolic cost for the prescribed step width control
condition was 3.1870.77 W/kg. In the preferred step
width control condition, subjects walked with a preferred step width of 0.12170.029L (Fig. 6). Foot
placement variabilities were 0.01670.003L (rms step
width) and 0.02670.009L (rms step length). Peak-topeak lateral displacement of the COM was
0.03370.008L. Net metabolic cost for the preferred
step width control condition was 2.9670.45 W/kg.
For the prescribed step width conditions, external
lateral stabilization reduced both foot placement variability and metabolic cost (Fig. 5). With external
stabilization, subjects walked with a step width of
0.01070.007L, and a step length of 0.75670.052L,
neither of which were significantly different from the
control trials (p ¼ 0:64 and p ¼ 0:78; respectively). In
contrast, foot placement variabilities decreased significantly, with a 33% decrease in rms step width to
0.01170.006L (p ¼ 3:1 104 ), and a 16% decrease in
rms step length to 0.02370.004L (p ¼ 4:8 103 ). The
reduction in step width variability was 201% greater
than the reduction in step length variability (p ¼ 0:016).
Peak to-peak lateral displacement of the COM was not
significantly different from the control condition
(0.00870.004L; p ¼ 0:22). External stabilization resulted in a 9.2% decrease in the metabolic cost of
prescribed step width walking, to 2.8270.41 W/kg
(p ¼ 0:025).
For the preferred step width conditions, external
lateral stabilization reduced preferred step width, foot
ARTICLE IN PRESS
J.M. Donelan et al. / Journal of Biomechanics 37 (2004) 827–835
Condition: Preferred Step Width
Metabolic cost
0.04
0.00
0.00
(a)
0.2
2
0.1
1
0
0.0
0.3
0.12
0.12
0.08
0.08
0.04
0.04
0.00
0.00
(b)
0.00
0.00
Step width
rms
Step length
rms
Fig. 5. Effects of external lateral stabilization on subjects walking at
the prescribed zero step width. (a) Average step width was unaffected
by the external lateral stabilizer and was close to the target zero width.
External stabilization resulted in decreases in (b) metabolic cost, and
(c) step width and length variability. Values shown are means7s.d.,
N ¼ 10: Asterisk (), denotes a statistically significant difference
(po0.05).
Distance (L)
0.01
0.03
0.03
Distance (m)
Distance (L)
(c)
0.01
0
0.0
Step width and length variability
0.03
0.02
1
0.04
0.04
*
0.1
with external stabilization
Step width and length variability
0.02
2
without external stabilization
with external stabilization
*
3
0.2
(b)
(a)
without external stabilization
0.03
*
0.02
*
0.02
Distance (m)
0.04
3
4
*
Cost of transport
0.08
Distance (m)
Distance (L)
0.08
Cost of transport
0.3
0.12
0.12
4
*
Metabolic cost
0.16
Distance (L)
0.16
Net metabolic power (W/kg)
Average step width
Average step width
Net metabolic power (W/kg)
Condition: Prescribed Step Width
Distance (m)
832
0.01
0.01
0.00
0.00
Step width
rms
Step length
rms
(c)
Fig. 6. Effects of external lateral stabilization on subjects walking at
their freely selected step width. External stabilization resulted in
decreases in (a) preferred step width (b) metabolic cost, and (c) step
width and length variability. Values shown are means7s.d., N ¼ 10:
Asterisk (), denotes a statistically significant difference (po0.05).
4. Discussion
placement variability and metabolic cost. Compared to
the control trials, subjects preferred a 47% narrower
step width of 0.06570.043L (p ¼ 3:4 104 ) when
externally stabilized (Fig. 6). No significant change
was observed in step length (p ¼ 0:55). Foot placement
variabilities decreased by 32% for rms step width and
14% for rms step length, to 0.01170.004L
(p ¼ 1:1105 ) and 0.02270.006L (p ¼ 0:055; not
statistically significant), respectively. The reduction in
step width variability was 235% greater than the
reduction in step length variability (p ¼ 0:036). Peakto-peak lateral displacement of the COM was
0.01470.008L, a 60% decrease from the control
condition (p ¼ 3:8 106 ). External stabilization also
resulted in a 5.7% reduction in the metabolic cost of
preferred step width walking, to 2.7870.38 W/kg
(p ¼ 0:015).
We provided external lateral stabilization to walking
subjects in order to understand the mechanical and
metabolic requirements of active lateral stabilization in
normal human walking. By reducing the active feedback
control required of the CNS (Fig. 1b), we addressed four
main questions. First, is walking passively unstable in
the lateral direction? If so, by what mechanism do we
actively stabilize? Does this active lateral stabilization
exact a significant metabolic cost? Finally, does passive
instability in the lateral direction affect our choice of
preferred step width? The observed reductions in foot
placement variability with external lateral stabilization—33% and 32% in prescribed and preferred step
width conditions, respectively—are in agreement with
our hypotheses that walking is passively unstable in the
lateral direction, requiring balance to be maintained
through active feedback control, largely by adjustments
in foot placement.
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Active lateral stabilization also appears to exact a
metabolic cost. Adjustments to foot placement are quite
small, typically only a few centimetres per step. Yet,
observed decreases in foot placement variability were
accompanied by modest but significant decreases in
metabolic cost. This is likely a result of two factors.
First, reduced variability implies reduced lateral
movement of the swing leg and this motion may
cost metabolic energy (Donelan et al., 2001; Shipman
et al., 2002). Second, reduced variability in medio-lateral
foot placement results in lower step-to-step transition
costs because transition costs are proportional to
the square of step width (Donelan et al., 2001). In the
freely selected step width conditions, part of
the energetic advantage might alternatively be due
to the narrower steps that subjects preferred when
laterally stabilized. Narrower steps require less mechanical work to redirect the COM between steps, with a
proportional reduction in metabolic cost (Donelan et al.,
2001, 2002). But this does not apply to the zero
step width conditions, which were designed to control
for step width, and external stabilization still resulted in
a 9.2% decrease in metabolic cost. Decreases in
foot placement variability are the most likely explanation for the decrease in metabolic cost with external
stabilization.
Our results also imply that lateral instability affects
the choice of preferred step width. The 43% decrease in
preferred step width with external stabilization indicates
that when there is no need to actively stabilize lateral
balance, humans take advantage of lower costs for stepto-step transitions at narrow step widths. External
stabilization presumably reduces the energetic costs that
would otherwise accompany narrow steps if active
control was necessary, resulting in a 5.7% decrease in
metabolic cost at the freely selected step width. There
remains, however, a greater metabolic cost at very
narrow step widths (3.18 W/kg at zero step width vs.
2.96 W/kg at preferred step width), regardless of
external stabilization. This prevents the preferred step
width from being zero, which would minimize step-tostep transition costs. The additional metabolic cost may
be for moving the swing limb laterally to avoid colliding
with the stance limb at very narrow step widths
(Donelan et al., 2001; Shipman et al., 2002). Another
disadvantage of very narrow steps is that they afford a
smaller margin of safety between the COM and center of
pressure. If the COM passes laterally outside the center
of pressure at any time during a step, large compensatory motions are usually needed to avoid falling. The
margin of safety may be quantified by the difference
between peak-to-peak lateral displacements of the COP
and COM. In the control conditions, this margin was
0.08870.023L for preferred width walking and
0.00670.009L for prescribed zero width walking. The
margin of safety for lateral COM motion therefore
833
decreases by 94% (p ¼ 3:9 107 ) when walking at zero
step width. The preferred step width appears to depend
on tradeoffs between active stabilization, step-to-step
transitions, and lateral limb motion.
The step width preferred by humans does not
minimize lateral motion of the COM. Minimization of
COM displacement in the vertical and lateral directions
has been hypothesized to reduce the energetic cost of
walking, with ‘‘six determinants of gait’’ that help to
achieve this minimization (Saunders et al., 1953).
However, we found that even in the control conditions,
the metabolic cost of walking increased by 7.4% when
walking with narrower steps even though subjects
reduced their lateral COM displacement by 85%.
Our mechanistic interpretation is that metabolic cost
is not determined by COM displacement per se, but
rather by the work performed to redirect the COM
(Donelan et al., 2001), to which COM displacement
bore a rough correlation. Factors, such as lateral
limb motion, also contribute to the higher metabolic
cost of narrow walking (without external stabilization),
making COM displacement a poor predictor of energy
expenditure.
Our conclusions are subject to several experimental
limitations. First, our protocol required subjects to walk
with their arms crossed, which may be less stable than
normal walking, perhaps magnifying the effect of
external stabilization. Second, components of the
external stabilizer unavoidably dissipate energy, thereby
increasing the positive work required of our subjects.
This disadvantage acts to reduce, rather than amplify,
the differences we observed. Third, the external stabilizer may produce unwanted forces in the vertical or
fore-aft directions. We estimate that these non-lateral
forces are small (see Section 2). Finally, the stabilizer
might reduce metabolic cost by providing a moment
about the hip that would act to support the trunk. To
prevent this, the stabilizer forces were applied with a line
of action passing very close to the hip. These limitations
may have a slight effect on our quantitative findings but
we feel that they are insufficient to affect our main
conclusions.
In addition to providing stabilizing forces, our
external lateral stabilizer may have a secondary effect
of providing subjects with additional sensory feedback
of lateral motion. The device applies forces to the waist
proportional to the lateral position of the COM.
Cutaneous receptors could potentially sense these
forces, increasing the information available to the
CNS for estimating COM motion. Just as reduced
vision results in increased step variability (Bauby and
Kuo, 2000), augmented feedback from lateral forces
could be partially responsible for the observed decreases
in step variability. However, if this were the case we
would have expected a learning effect as subjects gained
experience with these lateral forces and incorporated
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them into their control strategy. Instead, subjects
showed an immediate and consistent decline in step
variability and metabolic cost.
External lateral stabilization decreased both step
width and step length variability. We focused on step
width variability because medio-lateral foot placement is
an indicator of active control of lateral balance. But
some changes in step length variability are also to be
expected because in human walking there is a coupling
between medio-lateral and fore-aft motion (Bauby and
Kuo, 2000). Whether subjects were walking at the
prescribed or freely selected step width, the decreases in
step width variability exceeded those in step length
variability.
The ability to artificially provide lateral stability while
walking may have applications in rehabilitation medicine. Partial body weight support is commonly used for
patient groups lacking the strength to provide sufficient
vertical forces (e.g. Hesse, 2001). External lateral
stabilization could potentially be useful for patients that
have the strength to support their body weight, but have
difficulty controlling balance. By reducing the need for
active feedback control, external stabilization could
allow these patients to exercise, or to practice stabilizing
strategies and other aspects of gait mechanics, with
greater safety and confidence.
Acknowledgements
This research was supported in part by an NSERC
fellowship to J.M. Donelan, NIH grant AR44688 to R.
Kram and NIH grant DC0231201A1 to A.D. Kuo.
These experiments were conducted at the University of
California, Berkeley.
In this equation, xk is the state of the system at the
beginning of the kth step, and xkþ1 is the state at the
beginning of the next step. A passive walking gait has
states x (termed fixed points) for which, apart from
switching support from the left leg to the right, each step
is identical to the preceding one
x ¼ f ðx Þ:
ðA:2Þ
Passive dynamic walking gaits typically have two limit
cycles called long and short period gaits due to their
difference in step period. We have restricted our study to
the long period gait as it is most similar to human
walking (McGeer, 1990).
Stability is examined by evaluating the effect of small
deviations about the fixed point, using a linearized
approximation of Eq. (A.2)
ðx
x ÞEJðx Þðx x Þ;
ðA:3Þ
kþ1
where
Jðx Þ ¼
k
qf
:
qxk x
ðA:4Þ
The eigenvalues of the Jacobian matrix Jðx Þ
determine whether or not a perturbation (xk 2x ) will
be amplified or attenuated by the beginning of the next
step. If it is amplified, the limit cycle is unstable.
Conversely, if the perturbation is attenuated by the end
of the first step, it will continue to shrink in subsequent
steps—the system is locally stable and the passive
dynamic walker will return to its periodic gait. The
complex-valued eigenvalues must all be within the unit
circle for the system to be stable. The corresponding
eigenvectors describe modes along which the perturbations are decoupled from each other.
References
Appendix A
This appendix summarizes our stability analysis of the
passive dynamic walking model. These methods are
standard techniques for determining stability in nonlinear dynamical systems (Strogatz, 1994), first applied
to walking by McGeer (1990).
A step by a passive dynamic walker is divided into
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swing leg contacts the ground. This is followed by
double support, modeled as an instantaneous, inelastic
collision returning initial conditions for the next single
support phase. These two phases are combined to form
a vector-valued non-linear difference equation with the
step-to-step function, f
xkþ1 ¼ f ðxk Þ:
ðA:1Þ
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